Since risky positions in multivariate portfolios can be offset by various choices of
capital requirements that depend on the exchange rules and related transaction costs, it
is natural to assume that the risk measures of random vectors are set-valued.
Furthermore, it is reasonable to include the exchange rules in the argument of the risk
and so consider risk measures of set-valued portfolios. This situation includes the
classical Kabanov's transaction costs model, where the set-valued portfolio is given by
the sum of a random vector and an exchange cone, but also a number of further cases of
additional liquidity constraints.
The definition of the selection risk measure is based on calling a set-valued portfolio
acceptable if it possesses a selection with all individually acceptable marginals. The
obtained risk measure is coherent (or convex), law invariant and has values being upper
convex closed sets. We describe the dual representation of the selection risk measure
and suggest efficient ways of approximating it from below and from above. In case of
Kabanov's exchange cone model, it is shown how the selection risk measure relates to
the set-valued risk measures considered by Kulikov (2008), Hamel and Heyde (2010),
and Hamel et al. (2013)Supported by the Spanish Ministry of Science and Innovation Grants No. MTM20II—22993 and ECO20ll-25706. Supported by the Chair of Excellence Programme of the Universidad Carlos III de Madrid and Banco
Santander and the Swiss National Foundation Grant No. 200021-13752
